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The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichm¨uller Theory By Shinichi Mochizuki
Received xxxx xx, 2016. Revised xxxx xx, 2020
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Contents
§ 2. Changes of universe as arithmetic changes of coordinates
§ 2.1. The issue of bounding heights: the ABC and Szpiro Conjectures
A brief exposition of various conjectures related to this issue of bounding heights of rational points may be found in [Fsk], §1.3. In this context, the case where the algebraic
curve under consideration is the projective line minus three points corresponds most
directly to the so-called ABC and － by thinking of this projective line as the “λ-line”
that appears in discussions of the Legendre form of the Weierstrass equation for an
elliptic curve － Szpiro Conjectures. In this case, the height of a rational point may
be thought of as a suitable weighted sum of the valuations of the q-parameters of
the elliptic curve determined by the rational point at the nonarchimedean primes of potentially multiplicative reduction [cf. the discussion at the end of [Fsk], §2.2; [GenEll],
Proposition 3.4]. Here, it is also useful to recall [cf. [GenEll], Theorem 2.1] that, in the
situation of the ABC or Szpiro Conjectures, one may assume, without loss of generality,
that, for any given finite set Σ of [archimedean and nonarchimedean] valuations of the
rational number field Q,
the rational points under consideration lie, at each valuation of Σ, inside some
compact subset [i.e., of the set of rational points of the projective line minus
three points over some finite extension of the completion of Q at this valuation]
satisfying certain properties.
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